1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Urgent help: Euler Question

  1. Oct 12, 2006 #1

    How do I solve the equation x^7 \equiv 21 modulo 66

    66 = 2.3.11 so try solving it mod 3 and mod 11 (mod 2 doesn't gives any new information). This tells us x is divisible mod 3, and x^7 = -1 mod 11. One solution to this is x = -1 mod 7. The smallest solution of these two congruences is x=21, so this seems like a good guess.

    Now notice that 21*21 = 3.7 (2.11 - 1) = 7.66 - 3.7 = -21 mod 66
    So 21 ^n = 21. (-1)^n mod 66. Therefore x=21 is a solution of the equation. There may be more however...

    Sincerely Yours
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted